Fell in love but became discouraged (4,5)
I believe the answer is:
lost heart
'became discouraged' is the definition.
(I've seen this in another clue)
'fell in love' is the wordplay.
'fell in' becomes 'lost' (I can't explain this - if you can you should give a lot more credence to this answer).
'love' becomes 'heart' (heart is a kind of love).
'lost'+'heart'='LOST HEART'
'but' acts as a link.
(Other definitions for lost heart that I've seen before include "Despaired" , "disillusioned" , "Gave up" .)